Approximation for absolutely continuous functions by Stancu Beta operators
In this paper, we obtain an exact estimate for the first-order absolute moment of Stancu Beta operators by means of the Stirling formula and integral operations. Then we use this estimate for establishing a theorem on approximation of absolutely continuous functions by Stancu Beta operators.
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| author | Zeng, Xiao Ming Цзен, Сяо Мін |
| author_facet | Zeng, Xiao Ming Цзен, Сяо Мін |
| author_sort | Zeng, Xiao Ming |
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| description | In this paper, we obtain an exact estimate for the first-order absolute moment of Stancu Beta operators by means of the Stirling formula and integral operations.
Then we use this estimate for establishing a theorem on approximation of absolutely continuous functions by Stancu Beta operators. |
| first_indexed | 2026-03-24T02:31:06Z |
| format | Article |
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UDC 517.5
Xiao-Ming Zeng (Xiamen Univ., China)
APPROXIMATION FOR ABSOLUTELY CONTINUOUS
FUNCTIONS BY STANCU BETA OPERATORS *
НАБЛИЖЕННЯ АБСОЛЮТНО НЕПЕРЕРВНИХ ФУНКЦIЙ
БЕТА-ОПЕРАТОРАМИ СТАНКУ
In this paper, we obtain an exact estimate for the first-order absolute moment of Stancu Beta operators by
means of the Stirling formula and integral operations. Then we use this estimate for establishing a theorem on
approximation of absolutely continuous functions by Stancu Beta operators.
Отримано точну оцiнку для абсолютного моменту бета-операторiв Станку першого порядку iз вико-
ристанням формули Стiрлiнга та iнтегральних операцiй. Цю оцiнку використано для встановлення
теореми про наближення абсолютно неперервних функцiй бета-операторами Станку.
1. Introduction and definitions. For Lebesgue integrable functions f on the interval
I = (0,∞), Stancu Beta operators Ln are defined by
Ln(f ;x) =
1
B(nx, n+ 1)
∞∫
0
tnx−1
(1 + t)nx+n+1
f(t)dt, (1)
where B(p, q) is Beta function. The operators Ln were introduced first by Stancu [1]
in 1995. Stancu [1] studied some approximation properties of these operators. Abel [2]
derived the complete asymptotic expansion for the sequence of these operators. Abel,
Gupta and others [3, 4] discussed the rates of convergence of the operators Ln for
functions of bounded variation and functions with derivatives of bounded variation. For
more related work, one can refer to [5 – 7]. In present paper we first give an exact
estimate for the first order absolute moment of the operators Ln, then by means of
this estimate we establish an approximation theorem of operators Ln for the absolutely
continuous functions f ∈ ΦDB . The class of functions ΦDB is defined as follows:
ΦDB =
f | f(x)− f(0) =
x∫
0
φ(t)dt;
φ is bounded on every finite subinterval of [0,∞);
and f(t) = O(tr) as t→∞
.
For a bounded function f on every finite subinterval of [0,∞), we introduce the follow-
ing metric form:
Ωx(f, λ) = sup
t∈[x−λ, x+λ]∩[0,∞]
|f(t)− f(x)|,
where x ∈ [0,∞) is fixed, λ ≥ 0.
For the properties of Ωx(f, λ), one can refer to reference [8].
*This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), the
Natural Science Foundation of Fujian Province of China (Grant No. 2010J01012) and the National Defense
Basic Scientific Research Program of China (Grant No. B1420110155).
c© XIAO-MING ZENG, 2011
1570 ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
APPROXIMATION FOR ABSOLUTELY CONTINUOUS FUNCTIONS . . . 1571
2. The first absolute moment of Stancu Beta operators. In this section we derive
an exact estimate for the first order absolute moment of Stancu Beta operators. We need
the following lemma.
Lemma 1 ([2], Proposition 2). For Stancu Beta operators Ln, we have
Ln(1;x) = 1, Ln(t;x) = x, Ln((t− x)2;x) =
x(1 + x)
n− 1
. (2)
Let x ∈ (0,∞) be fixed. For r = 0, 1, 2, . . . , there holds
Ln((t− x)r;x) = O(n−b(r+1)/2c). (3)
Theorem 1. For Stancu Beta operators Ln, we have
Ln(|t− x|;x) =
2xnx
n(1 + x)nx+nn!
Γ(nx+ n+ 1)
Γ(nx)
. (4)
Proof. By direct computation and using Lemma 1, we have
Ln(|t− x|;x) =
=
1
B(nx, n+ 1)
x∫
0
(x− t)tnx−1
(1 + t)nx+n+1
dt+
∞∫
x
(t− x)tnx−1
(1 + t)nx+n+1
dt
=
=
2
B(nx, n+ 1)
x∫
0
(x− t)tnx−1
(1 + t)nx+n+1
dt+ Ln((t− x);x) =
=
2
B(nx, n+ 1)
x∫
0
xtnx−1
(1 + t)nx+n+1
dt−
x∫
0
tnx
(1 + t)nx+n+1
dt
.
Change of variable and integration by parts derive
x∫
0
xtnx−1
(1 + t)nx+n+1
dt = x
x/(1+x)∫
0
unx−1(1− u)ndu =
=
(
x
1 + x
)nx(
1
(1 + x)
)n
n
+
x/(1+x)∫
0
unx(1− u)n−1du, (5)
and
x∫
0
tnx
(1 + t)nx+n+1
dt =
x/(1+x)∫
0
unx(1− u)n−1du. (6)
From (5), (6) and simple computation, we obtain
Ln (|t− x|;x) =
2xnx
n(1 + x)nx+nB(nx, n+ 1)
.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1572 XIAO-MING ZENG
Note that
B(nx, n+ 1) =
n!Γ(nx)
Γ(nx+ n+ 1)
,
we obtain (4).
Theorem 1 is proved.
From Theorem 1 we derive the following inequalities, which are suitable for actual
applications.
Proposition 1. For Stancu Beta operators Ln, we have(
2x(1 + x)
πn
)1/2(
1− 1 + x
12nx
)
< Ln(|t− x|;x) <
(
2x(1 + x)
πn
)1/2
. (7)
Proof. By Theorem 1 and using Stirling’s formula (cf. [9, 10])
Γ(z + 1) =
√
2πz(z/e)zeθ, (12z + 1)−1 < θ < (12z)−1,
we have
Ln(|t− x|;x) =
2xnx
n(1 + x)nx+nn!
Γ(nx+ n+ 1)
Γ(nx)
=
(
2x(1 + x)
πn
)1/2
eθ1−θ2−θ3 ,
where
1
12(nx+ n) + 1
< θ1 <
1
12(nx+ n)
,
1
12n+ 1
< θ2 <
1
12n
,
1
12nx+ 1
< θ3 <
1
12nx
.
Set c(θ) = θ1 − θ2 − θ3, simple computation derives
−1 + x
12nx
< c(θ) < 0.
Thus we obtain(
2x(1 + x)
πn
)1/2
e−(1+x)/(12nx) < Ln(|t− x|;x) <
(
2x(1 + x)
πn
)1/2
.
It follows that(
2x(1 + x)
πn
)1/2(
1− 1 + x
12nx
)
< Ln(|t− x|;x) <
(
2x(1 + x)
πn
)1/2
.
Proposition 1 is proved.
Corollary 1. For Stancu Beta operators Ln and every x ∈ (0,∞), there holds
Ln(|t− x|;x) =
(
2x(1 + x)
πn
)1/2
+O(n−3/2).
3. Approximation for absolutely continuous functions. In this section we study
the rate of convergence of Stancu Beta operators Ln for the functions f ∈ ΦDB . We
need following two lemmas.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
APPROXIMATION FOR ABSOLUTELY CONTINUOUS FUNCTIONS . . . 1573
Lemma 2 ([3], Lemma 3). Let x ∈ (0,∞) be fixed, the functions Kn,x and Rn,x
are defined by
Kn,x(t) =
1
B(nx, n+ 1)
t∫
0
unx−1
(1 + u)nx+n+1
du, (8)
Rn,x(t) = 1−Kn,x(t) =
1
B(nx, n+ 1)
∞∫
t
unx−1
(1 + u)nx+n+1
du. (9)
Then for n ≥ 2, we have
Kn,x(y) ≤ x(1 + x)
(n− 1)(x− y)2
, 0 ≤ y < x, (10)
Rn,x(z) = 1−Kn,x(z) ≤ x(1 + x)
(n− 1)(z − x)2
, x < z <∞. (11)
Lemma 3. Let r > 0 and m be a positive integer satisfying m > r/2, we have
1
B(nx, n+ 1)
∞∫
2x
tr
tnx−1
(1 + t)nx+n+1
dt =
(2x)r
x2m
O
(
n−bm+1/2c
)
. (12)
Proof. For m > r/2 and t ∈ [2x,∞)
d
dt
tr
(t− x)2m
=
tr−1(t− x)2m−1(rt− 2mt− rx)
(t− x)4m
< 0,
which implies that
tr
(t− x)2m
is monotone decreasing for t ∈ [2x,∞). Thus
1
B(nx, n+ 1)
∞∫
2x
tr
tnx−1
(1 + t)nx+n+1
dt ≤
≤ 1
B(nx, n+ 1)
∞∫
2x
(2x)r
x2m
(t− x)2m
tnx−1
(1 + t)nx+n+1
dt ≤
≤ (2x)r
x2m
Ln
(
(t− x)2m;x
)
.
Equation (12) now follows from Lemma 1.
Now we state the main result of this section.
Theorem 2. Let f be a function in ΦDB . If φ(x+) and φ(x−) exist at a fixed
point x ∈ (0,∞), write τ =
φ(x+)− φ(x−)
2
, then for n ≥ 2 we have∣∣∣∣∣Ln(f ;x)− f(x)− τ
(
2x(1 + x)
nπ
)1/2∣∣∣∣∣ ≤
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1574 XIAO-MING ZENG
≤ 3 + 7x
n− 1
[
√
n ]∑
k=1
Ωx(φx, x/k) +
(1 + x)3/2√
72xπ
n−3/2 +
(2x)r
x2m
O
(
n−b(m+1)/2c
)
, (13)
where m is a positive integer satisfying m > r/2, and
φx(t) =
φ(t)− φ(x+), x < t ≤ 1,
0, t = x,
φ(t)− φ(x−), 0 ≤ t < x.
(14)
Proof. By direct computation we find that
Ln(f ;x)− f(x) =
φ(x+)− φ(x−)
2
Ln(|t− x|;x)−An,x(φx)+
+Bn,x(φx) +Dn,x(φx), (15)
where
An,x(φx) =
x∫
0
x∫
t
φx(u)du
dtKn,x(t),
Bn,x(φx) =
2x∫
x
t∫
x
φx(u)du
dtKn,x(t),
Dn,x(φx) =
∞∫
2x
t∫
x
φx(u)du
dtKn(x, t),
and Kn,x(t) is defined in (8).
Integration by parts derives
An,x(φx) =
x∫
0
x∫
t
φx(u)du
dtKn,x(t) =
=
x∫
t
φx(u)duKn,x(t)
∣∣∣∣∣∣
x
0
+
x∫
0
Kn,x(t)φx(t)dt =
x∫
0
Kn,x(t)φx(t)dt =
=
x−x/
√
n∫
0
+
x∫
x−x/
√
n
Kn,x(t)φx(t)dt.
Note that Kn,x(t) ≤ 1 and φx(x) = 0, by monotonicity of Ωx (φx, λ) it follows that∣∣∣∣∣∣∣
x∫
x−x/
√
n
Kn,x(t)φx(t)dt
∣∣∣∣∣∣∣ ≤
x√
n
Ωx
(
φx,
x√
n
)
≤ 2x
n
[
√
n ]∑
k=1
Ωx(φx, x/k).
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
APPROXIMATION FOR ABSOLUTELY CONTINUOUS FUNCTIONS . . . 1575
On the other hand, by inequality (10) and using change of variable t = x − x/u, we
have ∣∣∣∣∣∣∣
x−x/
√
n∫
0
Kn,x(t)φx(t)dt
∣∣∣∣∣∣∣ ≤
x(1 + x)
n− 1
x−x/
√
n∫
0
Ωx(φx, x− t)
(x− t)2
dt =
=
1 + x
n− 1
√
n∫
1
Ωx(φx, x/u)du ≤ 1 + x
n− 1
[
√
n ]∑
k=1
Ωx(φx, x/k).
Thus, it follows that
|An,x(φx)| ≤ 1 + 3x
n− 1
[
√
n ]∑
k=1
Ωx(φx, x/k). (16)
Next we estimate |Bn,x(φx)|
Bn,x(φx) =
2x∫
x
t∫
x
φx(u)du
dtKn,x(t) = −
2x∫
x
t∫
x
φx(u)du
dtRn,x(t) =
= −
t∫
x
φx(u)du ·Rn,x(t)
∣∣∣∣∣∣
2x
x
+
2x∫
x
φx(t)Rn,x(t)dt =
= −
2x∫
x
φx(u)du ·Rn,x(2x) +
∫ 2x
x
φx(t)Rn,x(t)dt. (17)
By Lemma 2∣∣∣∣∣∣−
2x∫
x
φx(u)du ·Rn,x(2x)
∣∣∣∣∣∣ ≤ xΩx(φx, x)
x(1 + x)
(n− 1)x2
=
(1 + x)
n− 1
Ωx(φx, x). (18)
On the other hand, similar to the estimate of |An,x(φx)|, we have∣∣∣∣∣∣
2x∫
x
φx(t)Rn(x, t)dt
∣∣∣∣∣∣ ≤ 1 + 3x
n− 1
[
√
n ]∑
k=1
Ωx(φx, x/k).
For estimate of |Dn,x(φx)|, note that f(t) = O(tr), thus there exists a constant M such
that
|Dn,x(φx)| ≤M
∞∫
2x
tr
tnx−1
(1 + t)nx+n+1
dt.
Using Lemma 3 we obtain
|Dn,x(φx)| = (2x)r
x2m
O(n−b(m+1)/2c). (19)
Theorem 2 now follows from (15) – (19) combining with Proposition 1 and some simple
computations.
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
1576 XIAO-MING ZENG
Remark. If f is a function with derivative of bounded variation, then f ∈ ΦDB .
Thus the approximation of functions with derivatives of bounded variation is a special
case of Theorem 2. In this special case Theorem 2 is better than a result of Gupta, Abel
and Ivan in [4].
1. Stancu D. D. On the Beta approximation operators of second kind // Rev. Anal. Numér. Théor. Approxim.
– 1995. – 2. – P. 231 – 239.
2. Abel U. Asymptotic approximation with Stancu Beta operators // Rev. Anal. Numér. Théor. Approxim.
– 1998. – 27. – P. 5 – 13.
3. Abel U., Gupta V. Rate of convergence of Stancu Beta operators for functions of bounded variation //
Rev. Anal. Numér. Théor. Approxim. – 2004. – 33. – P. 3 – 9.
4. Gupta V., Abel U., Ivan M. Rate of convergence of Beta operators of second kind for functions with
derivatives of bounded variation // Int. J. Math. and Math. Sci. – 2005. – 23. – P. 3827 – 3833.
5. Bojanic R., Cheng F. Rate of convergence of Bernstein polynomials for functions with derivatives of
bounded variation // J. Math. Anal. and Appl. – 1989. – 141. – P. 136 – 151.
6. Cheng F. On the rate of convergence of Bernstein polynomials of functions of bounded variation // J.
Approxim. Theory. – 1983. – 39. – P. 259 – 274.
7. Pych-Taberska P. Some properties of the Bézier – Kantorovich type operators // J. Approxim. Theory. –
2003. – 123. – P. 256 – 269.
8. Zeng X. M. Approximation properties of Gamma operators // J. Math. Anal. and Appl. – 2005. – 311. –
P. 389 – 401.
9. Mermin N. D. Stiring’s formula // Amer. J. Phys. – 1984. – 52. – P. 362 – 365.
10. Namias V. A simple derivation of Stiring’s asymptotic series // Amer. Math. Monthly. – 1986. – 93. –
P. 25 – 29.
Received 17.03.11
ISSN 1027-3190. Укр. мат. журн., 2011, т. 63, № 11
|
| id | umjimathkievua-article-2827 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:31:06Z |
| publishDate | 2011 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/03/9d2cd71889b414ce592633218c4e7603.pdf |
| spelling | umjimathkievua-article-28272020-03-18T19:37:24Z Approximation for absolutely continuous functions by Stancu Beta operators Наближення абсолютно неперервних функцiй бета-операторами станку Zeng, Xiao Ming Цзен, Сяо Мін In this paper, we obtain an exact estimate for the first-order absolute moment of Stancu Beta operators by means of the Stirling formula and integral operations. Then we use this estimate for establishing a theorem on approximation of absolutely continuous functions by Stancu Beta operators. Отримано точну оцiнку для абсолютного моменту бета-операторiв Станку першого порядку iз використанням формули Стiрлiнга та iнтегральних операцiй. Цю оцiнку використано для встановлення теореми про наближення абсолютно неперервних функцiй бета-операторами Станку. Institute of Mathematics, NAS of Ukraine 2011-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2827 Ukrains’kyi Matematychnyi Zhurnal; Vol. 63 No. 11 (2011); 1570-1576 Український математичний журнал; Том 63 № 11 (2011); 1570-1576 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2827/2407 https://umj.imath.kiev.ua/index.php/umj/article/view/2827/2408 Copyright (c) 2011 Zeng Xiao Ming |
| spellingShingle | Zeng, Xiao Ming Цзен, Сяо Мін Approximation for absolutely continuous functions by Stancu Beta operators |
| title | Approximation for absolutely continuous functions by Stancu Beta operators |
| title_alt | Наближення абсолютно неперервних функцiй бета-операторами станку |
| title_full | Approximation for absolutely continuous functions by Stancu Beta operators |
| title_fullStr | Approximation for absolutely continuous functions by Stancu Beta operators |
| title_full_unstemmed | Approximation for absolutely continuous functions by Stancu Beta operators |
| title_short | Approximation for absolutely continuous functions by Stancu Beta operators |
| title_sort | approximation for absolutely continuous functions by stancu beta operators |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2827 |
| work_keys_str_mv | AT zengxiaoming approximationforabsolutelycontinuousfunctionsbystancubetaoperators AT czensâomín approximationforabsolutelycontinuousfunctionsbystancubetaoperators AT zengxiaoming nabližennâabsolûtnoneperervnihfunkcijbetaoperatoramistanku AT czensâomín nabližennâabsolûtnoneperervnihfunkcijbetaoperatoramistanku |